Applying the Leray projector to (REF ) and noting that the Leray projector is bounded on \(L^p\)  when \(p \in (1, \infty )\)  (see, e.g., [1]}), we obtain (REF ). As for (), we take the curl of (REF ) and have
\(& \nabla ^\perp \cdot u = - \nabla ^\perp \rho \cdot \nabla \Phi \quad \text{if}\ d = 2,\\& \operatorname{\mathrm {curl}}u = - \nabla \rho \times \nabla \Phi \quad \text{if}\ d = 3.\) 
Then the proof is completed by invoking the well-known estimate
\(\Vert \nabla u\Vert _{L^r} \le C \Vert \operatorname{\mathrm {curl}}u\Vert _{L^r}\) 
for all \(r \in (1, \infty )\) .
